Researchers and teachers working together to deal with the issues

Researchers and teachers working together to deal with the issues

ZDM 2004 Vol. 36 (1)
Researchers and teachers working
together to deal with the issues,
opportunities and challenges of
implementing CAS into the senior
secondary mathematics classroom
Lynda Ball, Melbourne (Australia)
Abstract: Introducing CAS calculators into the mathematics
classroom has the potential to confront classroom practice more
than the introduction of graphics calculators. CAS calculators
are capable of performing many of the routine procedures that
are taught in the secondary school curriculum and having CAS
calculators available in the classroom has implications for pedagogy, interactions in the classroom and practice of routine
procedures. This paper will describe the way that the researchers and a group of teachers in three schools worked together to deal with issues and opportunities associated with the
implementation of CAS calculators into the year 11 and 12 curriculum during 2001 and 2002 in Victoria, Australia.
Kurzreferat: Kooperation von Forscher(inne)n und Lehrer(inne)n bezüglich des Umgangs mit Möglichkeiten und
Herausforderungen bei der Implementierung von CAS in der
Sekundarstufe. Die Einführung von CAS Rechnern in den
Mathematikunterricht fordert die unterrichtliche Praxis mehr als
jene von grafikfähigen Taschenrechnern. CAS Rechner sind in
der Lage viele Routineverfahren, die in der Sekundarstufe unterrichtet werden, auszuführen und die Verfügbarkeit von CAS
Rechnern im Unterricht hat Auswirkungen auf die Pädagogik,
die Interaktionen im Klassenzimmer und die Übung von Routineverfahren. In diesem Beitrag wird eine Form der Zusammenarbeit von Forscher(inne)n und einer Gruppe von Lehrer(inne)n
dreier Schulen in den Jahren 2001 und 2002 in Victoria,
Australien, beschrieben, die eingesetzt wurde, um Aspekte und
Möglichkeiten, die mit der Implementierung von CAS Rechnern
in den Jahrgängen 11 und 12 verbunden sind, zu behandeln.
ZDM-Classification: B50, C69, D30, D40, U70
1. The CAS-CAT teachers
This paper discusses four teachers Ken, Lucy, Meg and
Neil from three schools involved in the CAS-CAT research project in Victoria, Australia from mid 2000 to
December 2002. In this project students could use CAS
calculators at all times, including for assessment. Each
school used a different brand of CAS calculator.
The teachers implemented a new pilot course, with the
three project schools being the only schools in the state to
offer the new course. Students’ results in year 12 exams
contributed to university entrance and as a consequence
this new course was very “high stakes” for students. It
was important that the teachers and students had
appropriate support to make the transition from graphics
calculators to CAS calculators as smooth as possible.
Stacey (2003) and CAS-CAT research project (n.d.)
provide further information about the research project.
Ken, Lucy, Meg and Neil taught the new year 11 course
in 2001 and the new year 12 course in 2002. Three other
Analyses
teachers from the project schools also taught the year 11
course, but did not teach the new year 12 course during
2002.
The four project teachers started with a range of skills
in the use of CAS and different experience in teaching
year 12 mathematics. Mathematical Methods (CAS), the
new course (VCAA 2001) had much similar content and a
similar structure to the existing course called Mathematical Methods (VCAA 1999) and hence experience in
teaching Mathematical Methods was helpful in teaching
the new course. Lucy, Meg and Neil had extensive
experience in teaching Mathematical Methods. Ken, although an experienced year 12 teacher, had not taught
this particular course before. In volunteering to be involved in the research project, Ken needed both to learn
how to use CAS and to teach new content.
Lucy was recognised as an expert CAS user in Victoria
and was experienced at teaching the existing course. So
she was familiar with the content and extremely knowledgeable about the capabilities of CAS. Moreover, Lucy
was teaching mathematics using a brand of CAS calculator with which she was very familiar.
Before the project, Meg had used graphics calculators
regularly and CAS calculators occasionally and she had
knowledge of some symbolic features that could be useful for teaching. For this project Meg had very kindly
agreed to use a brand of CAS calculator that she was not
familiar with and which was different to the graphics
calculator that her school used in previous years. As a
result Meg had to learn new syntax using an unfamiliar
CAS system. Neither Neil nor Ken had used CAS
calculators prior to involvement in the research project.
Essentially Meg, Neil and Ken had to learn how to use a
new CAS calculator for the research project.
A first priority for the researchers was to provide
enough support to ensure all project teachers had facility
with basic CAS features of their brand. This would then
lead into consideration of pedagogical issues once
teachers were skilled at basic CAS use. Pedagogical
issues and opportunities would also arise naturally as
teachers and students used CAS in the classroom. All
participants felt that it was important to share the experiences of the teachers as they learned to teach with CAS.
Following is a discussion of the support, both formal
and informal provided by the researchers to assist
teachers in the implementation of CAS calculators into
their teaching.
2. Support from researchers
The researchers and teachers had to consider CAS related
issues, such as learning syntax and idiosyncrasies of
different brands, as well as curriculum, assessment and
teaching issues. The teachers and researchers worked
together to address various issues and the researchers also
provided material to the teachers to assist in CAS implementation. The researchers provided CAS syntax assistance (troubleshooting any queries), material and CAS
advice for new and current topics, assessment tasks and
examples of new types of questions in person, in project
documents and on the project website. In addition, they
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Analyses
initiated discussion about pedagogical opportunities with
CAS. These discussions occurred during five formal
project meetings, as well as during interviews, classroom
observations, emails and phone calls.
Initially the researchers had to help teachers develop facility with basic CAS use, particularly Ken and Neil who
had not used CAS prior to involvement in the project.
This was done through workshops held during project
meetings, development of worksheets to show how to
navigate the CAS calculator and basic CAS skills worksheets. Some teachers, including Meg, also requested that
researchers demonstrate basic CAS skills in classrooms,
so that students and teachers could both learn these skills
together. Meg believed that this was helpful to students
and at the same time it assisted in the development of her
own CAS skills. In an interview, Meg said that she could
see the benefits of this approach to help other teachers in
her school develop their skills in the future. Meg requested that a researcher do some examples in her class
on how to use the CAS and also how to record written
solutions when CAS is performing intermediate steps:
“… because I’m in the class you’re effectively teaching me at
the same time as well [as the students] … ”
When the interviewer asked Meg for her view about the
best way to introduce CAS technology to the next cohort
of CAS teachers, she again asked for assistance in class:
“… Team teach with them [the new CAS teachers] … I really do
think you need time to sit down with someone else and work
out how to use the [CAS] calculator. As I’ve said before, you
might have a course in front of you and you might be doing
polynomials and you really need to understand how you can use
the calculator effectively to teach it … I think you just need
somebody to help you and the best way to do that is, I think, if
you could do that for me now and then hopefully in the future.
If someone else is doing it next year I could team teach with
them [the new CAS teacher] and come into their class and show
their class how to do things and then it would develop that
way.”
Meg believed that the demonstration of CAS features and
pedagogical use of CAS was the best way to learn how to
use CAS. For Meg, the time spent with the researchers
where CAS use was an integral part of each meeting was
beneficial in the development of her basic CAS skills and
her ability to explore alternative teaching approaches.
One challenge in the project was having three different
brands of CAS calculator as support material to help
teachers and students learn syntax had to be tailored to
each brand of CAS. Sometimes this requires merely
alteration of syntax, but for some procedures, it requires
distinctly different instrumented techniques.
Another challenge was that a textbook did not exist for
the new course. The non-CAS textbooks used in years 11
and 12 were written for the existing course where the
highest level of technology expected was a graphics
calculator. For the new course, students had to cover the
material in the existing course as well as additional material. The different approaches possible with CAS were
obviously not apparent in the textbooks and in many
cases the approaches demonstrated in these textbooks
may not have been the most appropriate when students
had CAS. Textbooks provided extensive practice of pen28
ZDM 2004 Vol. 36 (1)
and-paper techniques. The extent of practice of pen-andpaper techniques required in the new course would most
likely be different to that in the existing course. Teachers
and students needed to find a balance between use of
CAS and practice of pen–and-paper techniques. The researchers provided guidance on the extent of pen-andpaper techniques and the amount of practice that might be
useful. Teachers and students were provided with written
support material and supplementary material to the textbook to ensure full coverage of the course content. In
year 12, the material was produced as a student booklet
(see CAS-CAT research project, n.d.). Ken believed that
the course material written by the researchers encouraged
a different focus for his teaching and helped him develop
a new teaching style:
“I suppose really when we first started back in Year 11, the
notes we got that we were starting functions and we were really
encouraged to teach the global and then sort of hone on in,
come in, come in, smaller.”
Lucy acknowledged the importance placed on the textbook by students and teachers and the increased level of
confidence that the course support material provided:
“… and it’s helped to have the course support materials because
it’s just very comforting for the kids, and for us [the teachers],
to know that there’s a sort of a sequence that still follows pretty
similarly what we’ve done in the past [in Mathematical
Methods]. There are slightly different emphases.”
Lucy believed that the course support material provided a
reference point for the changes that were occurring in her
teaching. The support material served three purposes,
firstly to provide the additional course content not covered in the textbooks, secondly to highlight some of the
pedagogical possibilities when CAS is used for teaching
and learning and thirdly to act as a reference for selected
CAS techniques. The researchers and teachers discussed
the new teaching approaches at meetings, which in turn
made all of the team consider both the opportunities that
access to CAS afforded as well as the classroom reality of
implementing a new technology in a high stakes course.
This discussion was crucial so that teachers and researchers were aware of the successes in the classroom
and the often intriguing but sometimes frustrating nature
of CAS. Sharing the experience of implementing the new
course was particularly important for Meg and Neil, both
of whom were the only teachers implementing the new
course in their respective schools. These discussions also
helped to illustrate that each brand of CAS had different
strengths and challenges. On occasion, the researchers
would present problems that highlighted the idiosyncrasies of particular CAS. This was useful as teachers
could choose to use these examples in class with students
to promote discerning use of CAS and to stress the
importance of “Algebraic Insight” (Pierce & Stacey
2002) to monitor CAS outputs.
The project teachers were very willing to articulate
their experiences and learning in the CAS classroom and
this may have been due to the extent of contact with the
researchers over the duration of the project. The researchers had a number of meetings with the teachers and various researchers did classroom visits, providing advice as
concerns occurred. In addition, the teachers and research-
ZDM 2004 Vol. 36 (1)
Analyses
ers communicated on a very regular basis via email,
phone and face-to-face. For Ken, one of the most useful
things was the presence of the researchers and knowing
that there was always someone available to troubleshoot
CAS or course queries:
“… To know that there’s always someone there to ask questions.
… You’re there by e-mail. … I mean, to know that you’re still
supportive.”
Neil explained how discussions with a researcher had made him recognise that his practice was to allow students
to communicate using calculator syntax when discussing
problems in class but then insisting on mathematical
language in written records.
“… [The students] talk in calculator syntax to me, but they write
it in mathematical language. I hadn’t ever thought about that
before. It’s not very often that they would write [ ... ] the
calculator syntax ... .”
The discussion with the researcher was the point where
Neil recognised this aspect of his teaching practice. Such
insight and reflection about teaching practice was common in discussions between researchers and teachers.
Table 1: Foci for project meetings 2001/2002
Date
Mar
2001
One
day
Sept
2001
Two
days
Feb
2002
Two
days
Aug
2002
One
day
Main foci at meeting
Accreditation of course
Research: Data collection
Support required by teachers to
use CAS: What would be
helpful?
‘Solving equations’and
‘calcu-lus’with CAS –
responses to email discussion
and student worksheets
(written by researchers)
Curriculum content
Research: interview findings
and implications for teachers
Basic CAS skills workshops
RIPA (Ball & Stacey 2003)
A rubric for written records
Types of questions and exams
Points for examiners
Application tasks and analysis
tasks: Should there be advice
about CAS use?
Research: Data collection
Strategies to help students
develop algebraic expectation
Advice for students about
written records
Sequence of topics
CAS Viewscreen
Answer
sample
exam
questions with CAS
Algebraic expectation and
effective use of CAS
Research: Data collection
Balance between pen-andpaper techniques and CAS
Support for teachers from
researchers
Anecdotes about CAS use
Issue/challenge
Administration
Administration
Pedagogy
Professional
development
Pedagogy
Curriculum
Pedagogy/
Research
Pedagogy
Professional
development
Pedagogy
Assessment
Assessment
Pedagogy
Assessment
Administration
Pedagogy
Pedagogy
Pedagogy
Pedagogy
Assessment
Pedagogy
Administration
Pedagogy
Administration/
Pedagogical
Pedagogy
Nov
2002
One
day
Teacher Interviews
Report on student work
Application task
Algebraic expectation trends
Pen-and-paper techniques test
CAS goals questionnaire
Pedagogy
Student use of
CAS
All areas
Table 1 shows the main foci for project meetings. All
project meetings with teachers were one or two days in
length mainly due to the geographical spread of the
schools. A further reason for having the teachers and researchers together for at least one full day was that this
allowed for consideration of a number of pedagogical
issues associated with introducing CAS, rather than just a
focus on the administration which was a necessary requirement when organizing data collection in schools.
Table 1 shows a clear focus on pedagogical issues when
the researchers and teachers met, particularly in 2002
when teachers were eager to share their classroom experiences and discuss the issues that they were facing. The
meetings provided the opportunity for teachers to share
their successes and discuss methods for overcoming CAS
concerns. Each meeting had at least one discussion that
raised pedagogical issues. For example, in March 2001,
one discussion point was the ‘Solving equations’material
that the teachers had trialed in term one.
Lucy, in discussing new types of questions that are possible when CAS is available, shows how the material
provided by the researchers helped in raising issues for
her:
“I must admit some of the questions that [researcher] has set
have been good for me for thinking about what could be different and they’ve been helpful for me for getting my head around
what different questions you might ask now to replace the ones
that are trivialized.”
The two day September meeting in 2001 involved basic
CAS workshops and discussion of research findings
which were mainly concerned with teacher and student
interviews. The meeting provided extended time to discuss issues, successes and challenges during 2001. For a
project that was running over an extended period of time,
it was important to have time for extended discussion. It
was also important to be able to identify issues that were
common across the three schools as well as any issues
that were unique so that the researchers could provide any
required support in different ways. The meeting provided
considerable momentum for the project, reinforcing the
positive relationships that had developed. This two day
meeting was very important for all members of the team.
The inclusion of basic CAS workshops in the September
2001 meeting was extremely effective in ensuring that all
teachers felt confident in using a range of CAS features.
Most teachers had, by that stage, the class sets of calculators for about 6 months, but they were still not confident
users. For the teachers, developing confidence with CAS
was a more substantial task than the researchers had expected. For example in an interview in May of 2001, Joe,
a Year 11 project teacher expressed his anxiety in using
technology for teaching mathematics:
“I’m uncomfortable with technology, very uncomfortable …
right at the beginning I was having to say to [Lucy], How do I
29
Analyses
do this? What buttons do I press? … [I would have to] have it
[the button sequence] written down before I went into the room
and I’m still not comfortable with the calculator. I mean I didn’t
get on top of the last one [graphics calculator] and then I’ve got
this one [CAS], so I find it hard.”
The change in Joe’s ability to use the CAS calculator and
increased confidence in using CAS calculators was apparent over the two days of the meeting. The significance of
this breakthrough was evident when Joe subsequently
invited a researcher to visit his class for the first time.
The second two day meeting was held in February
2002. The focus of this meeting was quite different to the
previous one as the teachers had moved to a position
where they were quite confident with basic CAS use and
had started to consider pedagogical implications of CAS.
The topics of discussion had made a definite shift to include aspects of the research. For example, a research
instrument that had been used to assess students’ algebraic expectation (Ball, Stacey & Pierce 2001) prompted
discussion about teaching approaches to help students
develop algebraic expectation. Another discussion concerned developing a new practice required for written
records of mathematical solutions when students have
CAS (see Ball & Stacey 2003). Having two solid days for
the team to work together was beneficial in helping provide a common direction and focus for the remainder of
2002.
The focus of meetings included discussion of issues
that had arisen from the research, planning for research
data collection, reporting on data and sharing of the
experience of implementing the CAS course for both
research purposes and to provide feedback to the course
developers and state education authority. The teachers
were very reflective about the pedagogical implications
of having CAS in their classrooms all of the time.
Through discussions at meetings the researchers and
teachers developed a common language for talking about
CAS use in the classroom. The discussions included consideration of research based findings and ways to improve student learning in the classroom.
Once teachers were familiar with basic CAS use they
could move their focus to developing students’ algebraic
expectation and consideration of issues such as pen-andpaper techniques versus CAS. Teachers also started to
have discussions about how to develop students’ understanding with CAS and the perceived impact on teaching
due to access to CAS. The teachers commented regularly
on how highly they valued the time that they had together
as a group of teachers to discuss pedagogy and experiences. The teachers said that their preference was to have
additional times to meet, rather than have to rely on email
or phone contact between project meetings, but
geographical distance made this difficult. The collegial
support provided by both the teacher group and the researchers was regarded as extremely important when implementing a new technology and particularly in considering teaching approaches for specific topics.
Teachers were also concerned with sharing the learning
from their involvement in the research project with other
teachers about to commence teaching with CAS. The
project teachers identified a number of issues that they
thought could be faced by teachers about to embark on
30
ZDM 2004 Vol. 36 (1)
teaching the new course. Some of these issues are shown
in Fig. 1.
•
•
•
•
•
•
•
Lack of textbook
Development of teacher and student expertise in using
CAS
Pedagogical implications of CAS
Including enough CAS in teaching
Not disadvantaging students.
Professional development requirements
Students developing proficiency with CAS
Fig. 1: Issues for teachers of the new course
Interestingly these issues demonstrate that the project
teachers fully recognised that CAS could have major
implications for pedagogy. Including a statement that
teachers must include enough CAS in the course reflects
the discussions held during the project. Some project
teachers were frequently commenting that they were concerned to include enough CAS in their teaching to ensure
that their students were competent in using CAS to solve
problems, particularly in year 12 courses where assessment is high stakes. Perhaps because of our strategy of
modifying an existing course and the necessity of using
existing textbooks, there was a tendency to overlook the
additional elements which CAS could provide in some
topics and to use it mainly to compensate for a few students weak algebraic skills. Teachers also stressed the
need for professional development, suggesting that in an
ideal situation teachers would be given the same level of
support that they received as project teachers.
3. The link between research and classroom practice
The project teachers gave a clear message at the final
meeting in November 2002. They believed that it was
important to share the experiences of the research project
and three of the teachers have presented at teacher conferences.
The following comment by Lucy demonstrates the high
level of reflection about teaching and learning with CAS
that has been articulated in many different ways by the
project teachers during the project:
“There should be a way of linking that [research outcomes] into
the PD [professional development] of the teachers because there
would be a real danger that it [professional development] would
become about button-pushing and learning the procedures and
all that. There’s so much more that’s come about, re-evaluating
what we do, the way that we do it, and the order that we do it,
and what things are still important and all that sort of stuff. So
the PD should focus on things like what’s still really important
and surely the outcomes of the research will indicate other
things that you want to make sure that are inculcated into teachers that are doing it. Because often you’ll get early adoptees that
are technology savvy and you actually need the input from the
teachers to say ‘Well these kids might be losing out’. You don’t
want to get lost in that argument but you want there to be a sort
of debate about what’s different about teaching in this environment. What things are the same, what things are different? And
what do we learn from the graphics calculator experiences that
we can mirror in the other [with CAS].”
ZDM 2004 Vol. 36 (1)
4. Conclusion
The teachers involved in this research project worked
very closely with the researchers over an extended period
of time. Crucial to this commitment to an extended research project was the availability of appropriate support
for the teachers. The project provided five forms of support for teachers
•
•
•
•
•
written materials
professional development in basic CAS use
full day meetings of the entire team (researchers and
teachers)
time for teachers to meet to support each other
Informal support via email and phone calls and during classroom observations
The researchers conducted research in the classrooms and
also provided teaching material and classroom based support. Importantly, the researchers were able to troubleshoot technical problems with the CAS calculators and
answer queries with regards to the new course. In the
initial stages, the technical support was of the highest
priority. As teachers developed their confidence and skills
in using the equipment in the classroom, their interest in
pedagogical issues came to the fore. As well as the support provided by the research team, the teachers also provided a large amount of support to each other. Towards
the end of the project, teachers’concerns moved towards
the professional development of others. The level of support by the researchers was high, but for an extended and
ambitious project it was necessary, both to support the
teachers and students and to maintain momentum for
pedagogical change.
Analyses
Stacey, K. (2003): Using computer algebra systems in secondary school mathematics: Issues of curriculum, assessment
and teaching. – In: W-C. Yang; S-C. Chu; T. de Alwis; M-G.
Lee (Eds.), Proceedings of the 8th Asian Technology Conference in Mathematics. USA: ATCM, p. 40-54.
Victorian Curriculum and Assessment Authority (2001): Units
1-4 Mathematical Methods (CAS) Pilot study. Retrieved 6
January 2004, http://www.vcaa.vic.edu.au.
Victorian Curriculum and Assessment Authority (1999): Mathematics Study Design. Melbourne: Author.
___________
Author
Ball, Lynda, Ms., Department of Science and Mathematics Education, The University of Melbourne, Victoria 3010,
Australia.
E-mail: lball@unimelb.edu.au
Acknowledgement
Many thanks to Kaye Stacey, the CAS-CAT team and project
school teachers. The CAS-CAT project was supported financially by the Australian Research Council, Hewlett-Packard
Australia Ltd, Shriro Australia (Casio), Texas Instruments and
the Victorian Curriculum and Assessment Authority Australia
Ltd.
References
Ball, L.; Stacey, K. (2003): What should students record when
solving problems with CAS? Reasons, information, the plan
and some answers. – In: J. T. Fey; A. Cuoco; C. Kieran; L.
Mullin; R. M. Zbiek (Eds.), Computer Algebra Systems in
Secondary School Mathematics Education. Reston, VA: The
National Council of Teachers of Mathematics, p. 289-303.
Ball, L.; Stacey, K. (in press): Teaching strategies for developing judicious technology use. – In: W. Masalski (Ed.),
Technology-Supported Mathematics Learning Environments
(National Council of Teachers of Mathematics Yearbook
2005). Reston, VA: NCTM.
Ball, L.; Stacey, K.; Pierce, R. (2001): Assessing algebraic expectation. – In: J. Bobis; B. Perry; M. Mitchelmore (Eds.),
Numeracy and Beyond. Proceedings of the 24th Annual Conference of the Mathematics Education Research Group of
Australasia. Sydney: MERGA, p. 66-73.
CAS-CAT Research Project (n.d.): Retrieved 22 December
2003, http://www.edfac.unimelb.edu.au/DSME/CAS-CAT.
Pierce R.; Stacey, K. (2002): Algebraic Insight: The algebra
needed to use computer algebra systems. – In: Mathematics
Teacher 95(8), p. 622-627.
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